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<h1>Modeling</h1>
<p>This model is used for simulating biofilm formation and the
stratification of concentration of oxygen</p>
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<h1>Introduction</h1>
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<p><strong>Compartment:</strong>The biofilm itself is distinguished
from the overlying water and the substratum to which it is attached. A
mass-transport boundary layer separates the biofilm from the overlying
water.</p>
<p>Within each compartment are<strong> components</strong>: include
different types of biomass ,substrates , products. biomass is often
divided into active microbial species, inert cells, and extracellular
polymeric substances(EPS).</p>
<p>The components can undergo transformation, transport, and
transfer <strong>processes</strong>. For example, substrate is consumed,
and this leads to the synthesis of new active biomass.</p>
<p>All process affecting each component in each compartment are
mathematically linked together into a<strong> mass balance
equation</strong> that contains rate terms and parameters for each process.</p>
<p><strong>Model Selection:</strong>Many kinds of Mathematics models
have been founded to describe a system of biofilm. Models of different
dimensions (1d, 2d, 3d) focus on different properties of a biofilm.
Since we care most about the oxygen concentration gradients
perpendicular to the substratum, <strong>numerical
1-dimensional dynamic model(N1)</strong> would be a proper choice for us.</p>


<h1>Compartment</h1>
<p><strong>The biofilm:</strong> A biofilm is a gel-like aggregation
of microorganisms and other particles embedded in extracellular
polymeric substancs. A biofilm contains water inside it, but its main
physical characteristic is that it is a solid phase. A biofilm normally
is anchored to a solid surface called the substratum on one side and in
contact with liquid on its other side. Frequently, a mass-transfer
boundary layer is included between the bulk liquid and the biofilm
itself. Thus, following figure illustrates a biofilm having four
compartments: the substratum, the biofilm itself, the boundary layer,
and the bulk liquid outside of the biofilm. While it is complex even for
a homogeneous biofilm morphology, we assume the biofilm surface is flat
and all material below the maximum biofilm thickness as part of the
biofilm components, and they have a constant density.</p>
<img src="http://igem.org/wiki/images/0/02/Modeling-biofilm.jpg"
	alt="biofilm" width="541" height="317" />
<p><strong>The mass-transport boundary layer: </strong>Experimental
observations clearly indicate strong concentration gradients for solutes
just outside the biofilm when these solutes are utilized or produced by
the microorganisms in the biofilm. Consequently, the solute
concentrations at the biofilm surface and in a completely mixed bulk
liquid often are significantly different.So we introduce the
mass-transport boundary layer,which is a hypothetical layer of liquid
above the biofilm and in which all the resistance to mass transport of
dissolved components outside the biofilm occurs.</p>
<p><strong>The bulk liquid:</strong> In our experiments, the bulk
liquid is large compared to the biofilm. So the simplest way seems to
consider it as a boundary condition of the biofilm compartment and
specify the concentrations of dissolved. However, dissolved components
can exchange between the biofilm and the bulk liquid, and it has a
profound impact on the concentrations in the bulk liquid. Thus we
include the bulk liquid not only as a boundary condition, but also as a
separate, completely mixed compartment, varying according to the inflow,
outflow, and the exchanges with the biofilm.</p>
<p><strong>The substratum:</strong> In our basic model, the
substratum is a separate compartment and impermeable. So it does not
have much effect on the biofilm system. However in some bioreactor, the
substratum may be permeable, or include organic solids that are
biodegraded by attached microorganisms.</p>
<h1>Component</h1>
<p><strong>Dissolved components:</strong> There are two kind of
dissolved components in our model, one is oxygen, the other is
substrates where nutrients are inside. They are expressed by inflow
concentration of oxygen, inflow concentration of substrates and monod
half saturation constant for substrates, monod half saturation constant
for oxygen.Diffusion coefficient of oxygen and diffusion coefficient of
substrate are also used to characterize the property of these
components.</p>
<p><strong>Particulate components:</strong> In our model, the
particulate components are microbes and EPS. We assume that they are
homogeneously mixed in the same proportion in all parts of the biofilm.
So can use bacterial density to relate the amount of bacterial and the
volume it takes up. And the volume fraction of EPS in Bacterial is
specified by a constant.</p>

<h1>Process</h1>
<p><strong>Transformation processes</strong> usually are biochemical
reactions that produce or consume one or more components: e.g.,
consumption of substrate, production of metabolic end-products,
microbial growth and decay, and production of EPS.</p>
<p>The<strong> transport processes</strong> that regularly are
considered in biofilm models are advection, molecular diffusion, and
turbulent dispersion. In special cases, transport of charged components
by migration in an electric field created is included. The general, 1d
expression to model the specific mass flux of a component is calculuted
in the direction z</p>
<p><strong>Transfer processes</strong> exchange mass of dissolved or
particulate components between two compartments. At the interface
between the compartments, a continuity condition for the component
concentration C and the specific flux j of the exchanged mass must be
fulfilled. Continuity means that C and j are the same on both sides of
the interface between the compartments. C and j can be calculated at
each side of the interface from boundary conditions</p>
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<h1>Modeling</h1>
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<a name="jumpmodeling">&nbsp;</a>
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<h1>An introduction to N1 model</h1>
<p>The core of the N1 model consists of a system of stiff,
non-linear partial differential equations. Because of the stiffness of
the equation system, integration methods tailored for stiff systems must
be used, or the dissolved and the particulate components have to be
treated differently. Furthermore, biofilm growth, i.e., the displacement
of the interface between biofilm and bulk liquid, creates a so-called
moving boundary problem (Kissel et al. 1984; Wanner and Gujer 1986).We
performed simulations with the N1 model using the software package
AQUASIM (Reichert 1998a, 1998b), since it can handle stiff systems and
the moving boundary. The N1 model implemented in AQUASIM is described in
next Section.</p>
<p></p>
<p><strong>The processes considered in the model include:</strong></p>
<ul>
	<li>Many transformation processes</li>
	<li>Advection and diffusion of attached particulate components in
	the biofilm solid matrix</li>
	<li>Attachment and detachment of particulate components at the
	biofilm surface and in the biofilm interior</li>
	<li>Diffusion of suspended particulate and dissolved components in
	the biofilm liquid phase and in the liquid boundary layer at the
	biofilm surface</li>
	<li>Complete mixing of suspended particulate and dissolved
	components in the bulk liquid</li>
</ul>
<h1>Definitions and equations</h1>
<p>As discussed above, biofilms are multiphase systems.
Consequently, in the N1 model three different phases are distinguished.
The solid attached phase is made up by the particulate components, which
form the biofilm solid matrix. In the model of single species of
bacteria, the particulate components are assumed to be uniform, defined
as</p>
<p><img src="http://ung.igem.org/wiki/images/0/02/Function1.png"
	width="660" style="float: none;" alt="function1" /></p>
<p>where<strong> X_M</strong> is the concentration of the attached
component, <strong>ρ</strong> is its density , defined as the mass
divided by the volume of the cell or particle, and<strong> ε</strong> is
its volume fraction, defined as volume of the component per unit biofilm
volume. The porosity or biofilm pore volume fraction<strong> θ
</strong>is</p>
<p><img src="http://ung.igem.org/wiki/images/c/cc/Function2.png"
	width=660 style="float: none;" alt="function2" /></p>
<p>The pore volume is formed by two phases: the phase of the
suspended particulate components with concentrations<strong>
X_P </strong>and the biofilm liquid phase, with the liquid phase volume fraction
<strong>ε_liquid</strong>:</p>
<p><img src="http://ung.igem.org/wiki/images/1/15/Function3.png"
	width="660" style="float: one;" alt="function3" /></p>

<h1>Mass balances in biobilm models</h1>
<p>Conservation of mass of a component in a dynamic and open system
states that:</p>
<p><img
	src="http://ung.igem.org/wiki/images/4/43/Function-massbalance.png"
	width="600" style="float: none;" alt="massbalance" /></p>
<p>The local mass balances are differential equations that express
the variation of concentration of a component in time in a point in
space as a result of transport and transformation processes.
One-dimensional mass balance equations for attached particulate,
suspended particulate, and dissolved components can be derived from the
general mass balance if gradients are considered only in the direction
of z, perpendicular to the substratum.</p>
<p><img src="http://ung.igem.org/wiki/images/0/00/Function4.png"
	width="660" style="float: none;" alt="function4" /></p>
<p>where<strong> t </strong>is time;<strong> z </strong>is
perpendicular coordinates ;<strong> S </strong>is the concentration;<strong>
j </strong>is the mass flux along the perpendicular coordinates; and<strong>
r </strong>is the net production rate of the component. For a dissolved substrate<strong>
S_i</strong>, the specific mass flux<strong> j_i </strong>in the biofilm, can be
modeled by Fick’s first law of diffusion as</p>
<p><img src="http://ung.igem.org/wiki/images/c/cc/Function5.png"
	width="660" style="float: none;" alt="function5" /></p>

<p>Where<strong> D_i </strong>is the molecular diffusivity of
substrate in the biofilm. Substituting this equation into the equation
(1) yields a mass balance equation that describes the development in
time and the spatial profile in the biofilm of the concentration<strong>
S_i </strong>of a dissolved substrate as</p>
<p><img src="http://ung.igem.org/wiki/images/d/da/Function6.png"
	width="660" style="float: none;" alt="function6" /></p>

<p>where<strong> r_i </strong>is the net production rate of<strong>
S_i</strong>. Equation (3) has the boundary condition</p>
<p><img src="http://ung.igem.org/wiki/images/7/78/Function7.png"
	width="660" style="float: none;" alt="function7" /></p>
<p>at the substratum (<strong>z=0</strong>) and</p>
<p><img src="http://ung.igem.org/wiki/images/2/22/Function8.png"
	width="660" style="float: none;" alt="function8" /></p>
<p>at the biofilm–bulk liquid interface (<strong>z=LF</strong>),
where LF is the biofilm thickness and <strong>S_LF</strong>, <strong>i</strong>
is the substrate concentration at the biofilm surface. All terms in
equation (6) are given as substrate mass per unit biofilm volume.
Equations (6) to (8) also hold for suspended particulate components,
i.e., for cells or particles which are suspended in the biofilm pore
volume and have the concentration<strong> X_P</strong>.</p>
<p>For attached particulate components, which form the biofilm solid
matrix, transport is assumed to be the result of microbial growth and
decay in the biofilm. Growing or shrinking cells lead to a volume
expansion or contraction of the biofilm solid matrix, respectively, and
to a displacement of neighboring cells (Wanner 1989). This displacement
can be interpreted as advective transport and is formally described as a
specific mass flux<strong> j_M </strong>by</p>
<p><img src="http://ung.igem.org/wiki/images/7/77/Function9.png"
	width="660" style="float: none;" alt="function9" /></p>

<p>where<strong> u_F </strong>is the distance by which the cells are
displaced per unit time. The displacement velocity<strong> u_F
</strong>of a cell at the location z is equal to the added net specific mass
production of the cells the biofilm matrix between the substratum and
this location:</p>
<p><img src="http://ung.igem.org/wiki/images/b/b0/Function10.png"
	width="660" style="float: none;" alt="function10" /></p>

<p>where<strong> r_M </strong>is the net production rate of a
particulate component in the biofilm matrix. Based on equations (9),
with<strong> u_F </strong>as defined in (10), a mass balance analogous
to equation (6) can be derived and used to describe the development in
time and the spatial profile in the biofilm of the attached particulate
component:</p>
<p><img src="http://ung.igem.org/wiki/images/9/98/Function11.png"
	width="660" style="float: none;" alt="function11" /></p>

<p>The boundary condition is no-flux condition:</p>
<p><img src="http://ung.igem.org/wiki/images/2/23/Function12.png"
	width="660" style="float: none;" alt="function12" /></p>

<p>at the substratum. Equation (11) is used to calculate the
relative abundance, spatial distribution, and development in time of
microbial species and particles in the biofilm.</p>
<p><img
	src="http://2011.igem.org/wiki/images/2/20/Biofilm-formaiton.jpg"
	width="660" style="float: none;" alt="biofilmformation" /></p>

<p>The development of the biofilm thickness in time is the result of
the net production of biomass in the biofilm, as described by equation
(10), of the attachment at the biofilm surface of microbial cells and
particles suspended in the bulk liquid, and of the detachment of
microbial cells and particles from the biofilm surface to the bulk
liquid. It is modeled as</p>
<p><img src="http://ung.igem.org/wiki/images/9/9c/Function13.png"
	style="float:none;s" width=660 margin-left=30px alt="function13" /></p>

<p>where<strong> u_de </strong>and<strong> u_at </strong>are the
detachment velocity and the attachment velocity, respectively. The
velocity<strong> u_de </strong>yields a phenomenological description of
the decrease of the biofilm thickness per unit time as result of the
detachment process, i.e., erosion or sloughing.</p>
<p>Attachment refers to the adsorption of microbial cells suspended
in the bulk liquid to the biofilm surface and is modeled by an
attachment velocity as</p>
<p><img src="http://ung.igem.org/wiki/images/9/97/Function14.png"
	width="660" style="float: none;" alt="function14" /></p>

<p>where <strong>k</strong> is the attachment rate coefficient and <strong>X_L</strong>
is the concentration at the bulk liquid side of the biofilm surface of
the suspended particulate component.</p>
<p>The environment of the biofilm is modeled as a completely mixed
volume of water, termed bulk liquid. Conversion processes in the bulk
liquid can be equally important as those that take place in the biofilm.
For each dissolved and particulate component considered, an additional
mass balance equation of the form</p>
<p><img src="http://ung.igem.org/wiki/images/4/4c/Function15.png"
	width="660" style="float: none;" alt="function15" /></p>

<p>is needed, where <strong>C_in</strong>,<strong>i</strong> and <strong>C_B</strong>,<strong>
i </strong>are the influent and bulk liquid concentrations, respectively, of the
dissolved or suspended particulate component<strong> i</strong>,<strong>
V_B </strong>is the bulk liquid volume,<strong> Q </strong>is the rate of flow
through the bulk liquid,<strong> A_F </strong>is the biofilm surface
area,<strong> j_F</strong>,i is the mass flux across the biofilm surface
, and<strong> r_B</strong>,<strong>i</strong> is the production rate of
the component.</p>
<h1>Transformation processes</h1>
<p>In modeling, the transformation processes are represented
mathematically with rate expressions, or equations that tell how fast
the component is produced or consumed. The rate is proportional to the
concentrations of one or more of the components and one or more kinetic
parameters. We describe the synthesis of microbial biomass with the
Monod equation for one limiting substrate:</p>
<p><img src="http://ung.igem.org/wiki/images/d/dc/Function16.png"
	style="float: none;" alt="function16" /></p>

<p>where<strong> μ </strong>is the specific growth rate, <strong>μ_max</strong>
the maximum specific growth rate,<strong> S </strong>is the
concentration of the rate-limiting substrate, and<strong> K_S </strong>is
the concentration giving one-half the maximum rate.</p>
<p>When multiple substrates are rate limiting, the Monod equation
typically is extended to include the effects of each substrate
influencing the rate of microbial synthesis. For instance, is the rate
of microbial synthesis may be limited by the concentrations of the
electron donor (S_1, e.g. COD) and the electron acceptor (<strong>S_2</strong>,
e.g. Oxygen) then the specific growth rate can be described with a
multiplicative-Monod expression (Bae and Rittmann 1996):</p>
<p><img src="http://ung.igem.org/wiki/images/6/68/Function17.png"
	width="660" style="float:none;"; alt="function17" /></p>

<p>Synthesis is not the only transformation process relevant to
microorganisms in a biofilm. Indeed, the microorganisms undergo a number
of loss processes, including endogenous decay (or inactivation),
predation, and detachment. The inactivation process is common in almost
any biofilm and serves as a model for how to express a biomass-loss
process.</p>
<p>Synthesis is not the only transformation process relevant to
microorganisms in a biofilm. Indeed, the microorganisms undergo a number
of loss processes, including endogenous decay (or inactivation),
predation, and detachment. The inactivation process is common in almost
any biofilm and serves as a model for how to express a biomass-loss
process.A simple and common means to represent inactivation/decay is
with a first-order loss rate</p>
<p><img src="http://ung.igem.org/wiki/images/9/95/Function18.png"
	width="660" style="float: none;" alt="function18" /></p>

<p>where<strong> r_in </strong>is the inactivation rate and b is the
first-order inactivation rate constant. Because utilization of a
substrate is what brings about biomass synthesis, the rate of substrate
utilization is proportional to the growth rate of<strong> X</strong>:</p>
<p><img src="http://ung.igem.org/wiki/images/a/a1/Function19.png"
	width="660" style="float: none;" alt="function19" /></p>
<p>where Y is the true-yield coefficient, or the ratio of biomass
produced per unit of substrate consumed. Similar equations could be
founded to describe rates in other processes. Following table summarizes
the basic rate expressions for heterotrophic bacteria in a convenient
and commonly used matrix format. The Monod half saturation constant for
oxygen<strong> K </strong>in processes of growth and respiration are
assumed to be equal.</p>
<p><img src="http://ung.igem.org/wiki/images/a/a7/Functions.png"
	width="660" alt="Functions"/></p>
</div>

<a name="mSimula">&nbsp;</a>
<div id="importantcontainer">
<div class="frame" id="important">
<div class="bgcolors" id="round">
<h1>Simulation</h1>
</div>
</div>
</div>

<div id="framecontent"><a name="jumpsimulation">&nbsp;</a>
<p>Calculate growth and oxygen concentration gradient of a biofilm
which consists of Heterotrophic bacteria. The water inflow at a constant
rate contains substrates and Oxygen. Growth occurs with Monod-type rate
laws. Respiration occurs with specific rate. <strong>A program
variable LF</strong> referring to Biofilm Thickness and <strong>3
dynamic volume state variables O_2, S, X</strong> referring to oxygen, substrates
and the Heterotrophic bacteria are defined. All necessary constants and
initial conditions are defined in following table. Some of their values
are based on average results provided by IWA.</p>
<p><img src="http://igem.org/wiki/images/d/d9/Zju-filmpara.png"
	style="width: 660px;" alt="peremater"/></p>
<h1>Definition of Process</h1>
<p>Define a dynamic process for growth, a dynamic process for
respiration, and a dynamic process for inactivation. The microbial
stoichiometric numbers of the processes of S (referring to substrates)
and O_2 (referring to Oxygen) and rate of the processes are shown in
following table.</p>
<p><img src="http://ung.igem.org/wiki/images/a/a7/Functions.png"
	width="660" alt="Functions"/></p>
<h1>Simulation results</h1>
<p><strong>Definition of simulation</strong></p>
<p>Define an active calculation with proper steps of 1 day.</p>

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    <div id="banner_bg"></div>  
    <div id="banner_info"></div> 
    <ul>
        <li class="on">1</li>
        <li>2</li>
   
    </ul>
   <div id="banner_list">
        <a href="#" target="_blank"><img src="http://2011.igem.org/wiki/images/3/38/Zju-results1.png" title="" alt="Biofilm thickness within two days 
" /></a>
      
        <a href="#" target="_blank"><img src="http://2011.igem.org/wiki/images/e/e5/Zju-results2.png" title="" alt="Biofilm thickness within six days" /></a>


    </div>
</div>
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		count=$("#banner_list1 a").length;
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		$("#banner1 li").click(function() {
			var i = $(this).text() - 1;//获取Li元素内的值，即1，2，3，4
			n1 = i;
			if (i >= count) return;
			$("#banner_info1").html($("#banner_list1 a").eq(i).find("img").attr('alt'));
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    {
        n1 = n1 >=(count - 1) ? 0 : ++n1;
        $("#banner1 li").eq(n1).trigger('click');
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</script>
<div id="banner1">    
    <div id="banner_bg"></div>  
    <div id="banner_info1"></div> 
    <ul>
        <li class="on">1</li>
        <li>2</li>
      
    </ul>
   <div id="banner_list1">
        <a href="#" target="_blank"><img src="http://2011.igem.org/wiki/images/c/cb/Zju-results3.png" title="" alt="Oxygen concentration in day 1
" /></a>
        
        <a href="#" target="_blank"><img src="http://2011.igem.org/wiki/images/8/80/Zju-results4.png" title="" alt="
Oxygen concentration in day 2" /></a>
    </div>
</div>
<!--
<img src="http://2011.igem.org/wiki/images/3/38/Zju-results1.png"
	alt="Results1"/> <img
	src="http://2011.igem.org/wiki/images/e/e5/Zju-results2.png"
	alt="Results2"/> <img
	src="http://2011.igem.org/wiki/images/c/cb/Zju-results3.png"
	alt="Results3"/> <img
	src="http://2011.igem.org/wiki/images/8/80/Zju-results4.png"
	alt="Results4"/> <img
	src="http://2011.igem.org/wiki/images/9/97/Zju-results5.png"
	alt="Results5"/>
-->
<p>Fit above graph into polynomial, we have:</p>
<img src="http://igem.org/wiki/images/e/e5/Zju-results6.png.png"
	alt="Results6"/>
</div>
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<h1>Reference</h1>
</div>
</div>
</div>
<div id="framecontent"><a name="mRefer">&nbsp;</a>
<table style="background-color: transparent;" width="0" border="0"
	cellspacing="1" cellpadding="1">
	<tr>
		<td>[1]</td>
		<td>O Wanner, H Eberl, E Morgenroth, D Noguera, C Picioreanu, B
		Rittmann, M Loosdrecht. ”Mathematical Modeling of Biofilms, IWA Task
		Group on Biofilm Modeling.”, Scientific and Technical Report, 2006.</td>
	</tr>
	<tr>
		<td>[2]</td>
		<td>P Reichert. “AQUASIM 2.0-Tutorial” Swiss Federal Institute
		for Environmental Science and Technology, 1998.</td>
	</tr>
	<tr>
		<td>[3]</td>
		<td>P Reichert. “AQUASIM 2.0-User Manual” Swiss Federal Institute
		for Environmental Science and Technology, 1998.</td>
	</tr>
	<tr>
		<td>[4]</td>
		<td>LI Tian-cheng, LI Xin-gang, ZHU Shen-lin.”Two-Dimension
		Dynamic Simulations on Biofilm Forming and Developing”. Acta
		Scientiarum Naturalium Universitatis Sunyatseni, 2005.</td>
	</tr>
</table>
<p></p>
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